Write as a Single Logarithm 2 log base 5 of 5x^3+1/3 log base 5 of x^2+6

$2 \log_{5} (5 x^{3}) + \frac{1}{3} \log_{5} (x^{2} + 6)$

Step 1

Simplify each term.

Simplify $2 \log_{5} (5 x^{3})$ by moving $2$ inside the logarithm.

$\log_{5} ({(5 x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Apply the product rule to $5 x^{3}$ .

$\log_{5} (5^{2} {(x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Rewrite $\log_{5} (5^{2} {(x^{3})}^{2})$ as $\log_{5} (5^{2}) + \log_{5} ({(x^{3})}^{2})$ .

$\log_{5} (5^{2}) + \log_{5} ({(x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Use logarithm rules to move $2$ out of the exponent.

$2 \log_{5} (5) + \log_{5} ({(x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Logarithm base $5$ of $5$ is $1$ .

$2 \cdot 1 + \log_{5} ({(x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Multiply $2$ by $1$ .

$2 + \log_{5} ({(x^{3})}^{2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Multiply the exponents in ${(x^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 + \log_{5} (x^{3 \cdot 2}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Multiply $3$ by $2$ .

$2 + \log_{5} (x^{6}) + \frac{1}{3} \cdot \log_{5} (x^{2} + 6)$

Simplify $\frac{1}{3} \log_{5} (x^{2} + 6)$ by moving $\frac{1}{3}$ inside the logarithm.

$2 + \log_{5} (x^{6}) + \log_{5} ({(x^{2} + 6)}^{\frac{1}{3}})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$2 + \log_{5} (x^{6} {(x^{2} + 6)}^{\frac{1}{3}})$